Not Principal Ideal

How would you prove it's not a P.I.D. by using I?

Let be an ideal of . Prove that I is not a principal ideal of .

I've been trying this for a long time... and I'm just saying okay it has this form, and then I multiply it by (a+bsqrt(-5)) and this happens... but I keep getting so many terms, that I can't find a contradiction. Please help!

I've been trying this for a long time... and I'm just saying okay it has this form, and then I multiply it by (a+bsqrt(-5)) and this happens... but I keep getting so many terms, that I can't find a contradiction. Please help!

Think about norms. , (in general, ).

You should therefore notice that if I is a principle ideal, the principle element (or what ever it is called - lets denote it by x) has norm 2 (why?). You should also notice that there exists some a and b such that and . So...can you find a contradiction in what I have just written?

I've been trying this for a long time... and I'm just saying okay it has this form, and then I multiply it by (a+bsqrt(-5)) and this happens... but I keep getting so many terms, that I can't find a contradiction. Please help!

Using, perhaps, the idea of norm Swlabr wrote (and you're right, it must be 5 and not 25. A

typo, surely) , prove that are two fundamentally

different factorizations of 6 in that ring and, thus, there is not unique factorization there, so